MACROSCOPIC MODELING AND SIMULATIONS OF
ROOM EVACUATION
arXiv:1308.1770v2 [math.NA] 14 Feb 2014
M. TWAROGOWSKA1 , P. GOATIN2 , AND R. DUVIGNEAU2
Abstract. We analyze numerically two macroscopic models of crowd
dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne-Whitham vehicular
traffic model. The desired direction of motion is determined by solving
an eikonal equation with density dependent running cost, which results
in minimization of the travel time and avoidance of congested areas. We
apply a mixed finite volume-finite element method to solve the problems
and present error analysis for the eikonal solver, gradient computation
and the second order model yielding a first order convergence. We show
that Hughes’ model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using
the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.
1. Introduction
Crowd dynamics has recently attracted the interests of a rapidly increasing number of scientists. Analytical and numerical analysis are effective
tools to investigate, predict and simulate complex behaviour of pedestrians,
and numerous engineering applications welcome the support of mathematical modelling. Growing population densities combined with easier transport
lead to greater accumulation of people and increase risk of life threatening
situations. Transport systems, sports events, holy sites or fire escapes are
just few examples where uncontrolled behaviour of a crowd may end up in
serious injuries and fatalities. In this field, pedestrian traffic management
is aimed at designing walking facilities which follow optimal requirements
regarding flow efficiency, pedestrians comfort and, above all, security and
safety.
From a mathematical point of view, a description of human crowds is
strongly non standard due to the intelligence and decision making abilities
of pedestrians. Their behaviour depends on the physical form of individuals and on the purpose and conditions of their motion. In particular,
pedestrians walk with the most comfortable speed [7], tend to maintain a
preferential direction towards their destination and avoid congested areas.
On the contrary, in life threatening circumstances, nervousness make them
1
Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle
Ricerche, via dei Taurini 19, I-00185 Roma, Italy (mtwarogowska@gmail.com).
2
INRIA Sophia Antipolis - Méditerranée, OPALE Project-Team, 2004, route des
Lucioles – BP 93, 06902 Sophia Antipolis Cedex, France (paola.goatin@inria.fr,
regis.duvigneau@inria.fr) .
keywords and phrases: macroscopic models, crowd dynamics, evacuation, Braess paradox
.
1
2
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
move faster [46], push others and follow the crowd instead of looking for
the optimal route [34]. As a consequence, critical crowd conditions appear
such as ”freezing by heating” and ”faster is slower” phenomena [22, 21],
stop-and-go waves, transition to irregular flow [24] and arching and clogging
at bottlenecks [46].
In order to describe this complex crowd dynamics, numerous mathematical models have been introduced, belonging to two fundamentally distinct approaches: microscopic and macroscopic. In the microscopic framework pedestrians are treated as individual entities whose trajectories are determined by physical and social laws. Examples of microscopic models are
the social force model [25], cellular automata models [41, 11], AI-based models [16]. Macroscopic description treats the crowd as a continuum medium
characterized by averaged quantities such as density and mean velocity. The
first modelling attempt is due to Hughes [29] who defined the crowd as a
”thinking fluid” and described the time evolution of its density using a scalar
conservation law. Current macroscopic models use gas dynamics equations
[2, 33], gradient flow methods [39], non linear conservation laws with non
classical shocks [8] and time evolving measures [45]. At an intermediate
level, kinetic models derive evolution equations for the probability distribution functions of macroscopic variables directly from microscopic interaction
laws between individuals, see for example [1] and [10] and references therein.
Also, recently introduced approaches include micro-macro coupling of timeevolving measures [9] and mean-field games [37]. These models are good
candidates to capture the effects of individual behavior on the whole system.
In this paper we shall analyze and compare two macroscopic models
describing the time evolution of the density of pedestrians. The first one,
introduced by Hughes [29], consists of a mass conservation equation supplemented with a phenomenological relation between the speed and the density
of pedestrians. The second one involves mass and momentum balance equations so is of second order type. It was proposed by Payne and Whitham
[44, 55] to describe vehicular traffic and adopted to describe pedestrian motion by Jiang et al. [33]. It consists of the two-dimensional Euler equations
with a relaxation source term. In both models, the pedestrians’ optimal
path is computed using the eikonal equation as was proposed by Hughes
[29].
In order to simulate realistic behaviour we consider two dimensional,
continuous walking domains with impenetrable walls and exits as pedestrians’ destination. To our knowledge the only available results using Hughes’
model concern simulations of flow of pedestrians on a large platform with
an obstacle in its interior [28, 32]. In the case of the second order model
Jiang et al. [33] considered the same setting and showed numerically the
formation of stop-and-go waves. However, none of the above works analyzed
complex crowd dynamics. Behaviour at bottlenecks and evacuation process
was not considered in any of the previous works.
The first aim of this paper is to provide a more detailed insight into
the properties of macroscopic models of pedestrian motion. In particular, we
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
3
compare Hughes’ model and the second order model analyzing the formation of stop-and-go waves and flows through bottlenecks. Our simulations
suggest that Hughes’ model is incapable of reproducing neither such waves
nor clogging at a narrow exit. It appears to be also insensitive to the presence of obstacles placed in the interior of the walking domain, which can be
crucial in the study of evacuation. This is why in the second part of the
paper we restrict ourselves only to the second order model
We focus on the study of the evacuation of pedestrians through a
narrow exit. This problem is an important safety issue because of arching
and clogging appearing in front of the exit, which can interrupt the outflow
and result in crushing of people under the pressure or the crowd. Experimental studies are rare due to the difficulties in reproducing realistic panic
behaviour [35, 50, 26], while numerical simulations are available mainly in
the microscopic framework. For example Helbing et al. [20, 23] analyzed
the evacuation of two hundred people from a room through a narrow door
and in [18] the issue of optimal design of walking facilities was addressed
with genetic algorithms. At first we show the dependence of the solutions
on different parameters of the model. More precisely, we consider the effect
on the evacuation of the strength of the interpersonal repealing forces and
the desired speed of pedestrians. Both of these parameters may indicate the
nervousness and the level of panic of pedestrians.
In order to improve evacuation, Hughes [30] suggested that suitably
placed obstacles can increase the flow through an exit. This idea is an
inversion of the Braess paradox [4, 5], which was formulated for traffic flows
and states that adding extra capacity to a network can in some cases reduce
the overall performance. In the case of crowd dynamics, placing an obstacle
may be seen intuitively as a worse condition. Nevertheless, it is expected to
lower the internal pressure between pedestrians and their density in front
of the exit and as a result preventing from clogging. This phenomenon has
been studied experimentally in case of granular materials by Zuriguel et al.
[56] who analyzed the outflow of grains from a silo and found out the optimal
height above the outlet of an obstacle which reduces the blocking of the flow
by a factor of one hundred. In case of pedestrians, to our knowledge, so far
this problem has been studied only numerically. Helbing et al. [21] using the
social force model observed that a single column placed in front of the exit
decreases the pressure between the column and the door and may prevent
from clogging. In the same framework, different shapes and placements of
obstacles were studied in [13] with an indication of the formation of the so
called ”waiting zone” in front of the exit. Frank and Dorso in [15] studied
the effects of a column and a longitudinal panel assuming in the social force
model that pedestrians change their direction away from an obstacle until
the exit becomes visible.
Following the idea of Hughes [30], we try to improve the evacuation of
pedestrians using properly tuned obstacles placed in front of the exit. Motivated by the numerical simulations in which clogging appears when a large
group of pedestrians reached the exit simultaneously, we give an example of
a system of five circular columns arranged in the shape of a triangle opened
towards the exit. We show that this system of obstacles effectively creates
4
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
an area with lower density in front of the door and reduces the clogging.
This paper is organized as follows: in Section 2 we explain in detail
macroscopic models and in Section 3 we describe numerical approximation
of the models. Section 4 is devoted to the numerical results. At first we
present error analysis and comparison between the two macroscopic models.
Then we analyze the evacuation of pedestrians from a room.
2. Macroscopic model of pedestrian flow
2.1. Equations. We consider a two dimensional connected domain Ω ⊂ R2
corresponding to some walking facility. It is equipped with an exit which
models the destination of the crowd motion and can contain obstacles. The
boundary of the domain Ω is composed of the outflow boundary Γo and the
wall Γw , which, as obstacles, is impenetrable for the pedestrians. In this
setting we consider a macroscopic model introduced by Payne-Whitham for
vehicular traffic flow in [44, 55] and by Jiang et al. in [33] to describe crowd
dynamics. The model derives from fluid dynamics and consists of mass and
momentum balance equations with source term. Denoting by ρ the density
of pedestrians and by ~v their mean velocity the model reads
(1)
ρt + div(ρ~v ) = 0,
~ ~v ),
(ρ~v )t + div(ρ~v ⊗ ~v ) = A(ρ,
~ ~v ) describes the average acceleration caused by internal driving
where A(ρ,
forces and motivations of pedestrians. More precisely, it consists of a relaxation term towards a desired velocity and the internal pressure preventing
from overcrowding
~ ~v ) = 1 (ρV (ρ)~
A(ρ,
µ − ρ~v ) − ∇P (ρ).
τ
The unit vector µ
~ = µ
~ (ρ(x, t)) describes the preferred direction pointing
the objective of the movement of pedestrians and will be defined in the
next section. The function V (ρ) characterizes how the speed of pedestrians
changes with density. Various speed-density relations are available in the
literature, see [7]. For our simulations we choose the exponential dependence
(2)
(3)
V (ρ) = vmax e
−α ρ
ρ
max
2
,
where vmax is a free flow speed, ρmax is a congestion density at which the
motion is hardly possible and α is a positive constant. The parameter τ in
(2) is a relaxation time describing how fast pedestrians correct their current
velocity to the desired one. The second term in (2) models a repulsive force
modeling the volume filling effect and is given by the power law for isentropic
gases
(4)
P (ρ) = p0 ργ ,
p0 > 0,
γ > 1.
Remark 2.1. Model (1) is referred to as a second order model as it consists
of mass and momentum balance equations completed with a phenomenological law describing the acceleration. A simpler system, which is a first order
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
5
model, was introduced by Hughes [29, 30]. It is composed of a scalar conservation law
(5)
ρt + divF~ (ρ) = 0,
where F~ (ρ) = ρV (ρ)~
µ, closed by a speed-density relation V (ρ) given by (3).
2.2. Desired velocity. The models (1), (5) have to be completed by defining the vector field µ
~ . Following the works of Hughes, we assume that the
pedestrians movement is opposite to the gradient of a scalar potential φ,
that is
∇φ
(6)
µ
~ =−
.
||∇φ||
The potential φ corresponds to an instantaneous travel cost which pedestrians want to minimize and is determined by the eikonal equation
|∇φ| = c(ρ) in Ω
(7)
,
φ=0
on Γo
where c(ρ) is a density dependent cost function increasing with ρ. In the
simplest case we could prescribe c(ρ) = 1, which gives the potential φ(x) =
dist(x, Γo ) in the case of convex domains. Pedestrians want to minimize
the path towards their destination but temper the estimated travel time by
avoiding high densities. The behaviour can be expressed by the ”density
driven” rearrangement of the equipotential curves of φ using the following
cost function [29]
(8)
c(ρ) =
1
.
V (ρ)
Remark 2.2. Instead of coupling the mass and momentum balance laws with
an eikonal equation, another possible approach has been recently introduced
in [39, 40]: the transport equation is interpreted as a gradient flow in the
Wasserstein space, which has the advantage of providing existence results
despite the non-smooth setting.
3. Numerical scheme
Let us now present the numerical scheme on unstructured triangular
mesh that we used to perform numerical simulations. The model of pedestrian flow couples equations of different nature, i.e. a two dimensional nonlinear system of conservation laws with sources, coupled with the eikonal
equation through the source term. In this section we describe a finite volume scheme built on dual cells for systems (1) and (5) and a finite element
method based on the variational principle for problem (7). The numerical
simulations are carried out using the multidisciplinary platform NUM3SIS
developed at Inria Sophia Antipolis [31, 36].
3.1. Finite volume schemes for the macroscopic models. The models
of pedestrian motion (1) and (5) can be put in the form
(9)
Ut + divF~ (U ) = S(U ),
6
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
where in the case of the second order model (1) U = (ρ, ρ~v )T denotes the
unknowns vector, density and momentum, and
ρ F
0
ρ~
v
F~ (U ) =
.
=
, S(U ) =
1
ρ~v ⊗ ~v + P (ρ)
µ − ρ~v )
F ρ~v
τ (ρV (ρ)~
For the first order model (5) we take U = ρ, F~ (U ) = ρV (ρ)~
µ and S(U ) = 0.
According to the framework of finite volume schemes, we decompose
the domain Ω into N non overlapping, finite volume cells Ci , i = 1, ..., N ,
given by dual cells centered at vertices of the triangular mesh. For each cell
Ci we consider a set of Ni neighbouring cells Cij , j = 1, ..., Ni . By eij we
denote the face between Ci and Cj , |eij | its length and ~nij is a unit vector
normal to the eij pointing from the center of the cell Ci towards the center
of the cell Cj . The solution U of the system (9) on a cell Ci is approximated
by the cell average of the solution at time t > 0, that is
Z
1
U (x, t)dx.
Ui =
|Ci | Ci
A general semi-discrete finite volume scheme for (9) can be defined as
N
(10)
i
d
1 X
|eij |F(Ui , Uj , ~nij ) + S(Ui ),
Ui = −
dt
|Ci |
j=1
where F(Ui , Uj , ~nij ) is a numerical flux function. The spatial discretization
of the source term S(Ui ) is treated by a pointwise approximation S(Ui ) =
(0, (ρi V (ρi )~
µi − ρi~vi ) /τ )T .
In order to obtain a numerical approximation using a finite volume
scheme (10) we have to compute numerical fluxes F(Ui , Uj , ~nij ) across the
face eij between control cells Ci and Cj along the direction ~nij . Despite the
fact that the model is two dimensional, these fluxes are computed using a
one-dimensional approximation.
The homogeneous part of the model (1) coincides with the isentropic
gas dynamics system for which many solvers are available, (see [52]). However, the occurrence of vacuum may cause instabilities and not all of them
preserve non negativity of the density. We use the first order HLL approximate Riemann solver [17]. It assumes that the solution consists of three
constant states separated by two waves with speeds σL and σR corresponding respectively to the slowest and fastest signal speeds. It is ”positivity
preserving” under certain conditions on the above numerical wave speeds
[12] that is
(11)
σL = min(vLn − sL , v̄Roe − s̄),
n + s ),
σR = max(v̄Roe + s̄, vR
R
p
where s = P 0 (ρ) is the sound speed, v n is the normal component of the
velocity and v̄Roe , s̄ are averaged Roe velocity and sound speed respectively.
For the numerical function in the case of the first order model we use
the Lax-Friedrichs flux
1
(12a)
F (ρi , ρj , ~nij ) = [F (ρi ) · ~nij + F (ρj ) · ~nij − ξ(ρj − ρi )] ,
2
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
7
with the numerical viscosity coefficient ξ given by
(12b)
d
d
d
ξ = max F (ρl ) = max (ρl V (ρl )~
µl (ρ)) = max (ρl V (ρl )) .
l=i,j dρ
l=i,j dρ
l=i,j dρ
The last equality is justified by the fact that µ
~ is a unit vector.
3.2. Fully discrete schemes. The difficulty in the time discretization of
equation (10) lies in the non linear coupling of the models with the eikonal
equation (7) in the flux for the first order model and in the source term
for the second order one. This is why we apply explicit time integration
method. Denoting the time step by ∆t, the density at the time step tn+1
is obtained by using an explicit Euler method with the splitting technique
between the transport and the source terms

Ni

 U ∗ = U n − ∆t X |e |F(U n , U n , ~n ),
ij
ij
i
j
i
i
(13)
|Ci |
j=1

 n+1
Ui
= Ui∗ + ∆tS(Ui∗ ),
where the numerical flux function F depends explicitly on µ
~ n . The stability is achieved under the CFL condition ∆t ≤ α · min diameter(Ci )/σi .
i=1,...,N
p
In case of the second order model σi = max (~vi · ~nij ) + P 0 (ρi ) is the
j=1,...,Ni
maximal value of the characteristic wave speed of the homogeneous part of
the system (1). For the first order model, using the speed-density
relation
d
(3), the maximal wave speed is given by σi = max F (ρ) = vmax due
i=1,...N dρ
to the same argument |~
µ| = 1 used for the computation of the coefficient ξ
in the Lax-Friedrichs flux (12). The value of α is set to 0.9 in the following
computations.
3.3. Solution to the eikonal equation and gradient. To obtain the
solution at time step tn+1 we need to compute the direction vector µ
~ defined
by (6). It means that we have to solve the eikonal equation (7) and compute
the gradient of its solution. Equation (7) is a special case of the static
Hamilton-Jacobi equation, for which many numerical methods have been
developed such as level-set methods [43, 42], fast marching and fast sweeping
methods [48, 53], semi-lagrangian scheme [14], finite volume or finite element
schemes [27, 6]. We implement the Bornemann and Rasch algorithm [3]
belonging to the last of the above approaches thus it is easier to implement
on unstructured triangular meshes with respect to other methods. It is a
linear, finite element discretization based on the solution to a simplified,
localized Dirichlet problem solved by the variational principle.
Having found the potential φ we calculate its gradient using the nodal
P1 Galerkin gradient method. It is related to cell Ci and is computed by
averaging the gradients of all triangles having node i as a vertex. In two
dimensions it has the form
1 X |Tij | X
φk ∇Pk |Tij ,
(14)
∇φi =
|Ci |
3
Tij ∈Ci
k∈Tij
8
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
Figure 1. Two dimensional domain Ωevac ∪ Ωempty = Ω ⊂ R2
with the boundary ∂Ω = Γwall ∩ Γoutf low . Initially the density is
positive only in the region Ω0
where Tij are triangles with the considered node i as a vertex, k counts for
vertices of Tij and Pk |Tij is a P1 basis function associated with vertex k.
3.4. Boundary and initial conditions. We perform simulations on a twodimensional domain Ω ⊂ R2 with boundary ∂Ω = Γo ∪ Γw , see Fig 1.
We set the outflow boundary Γo far from the exit of the room through
which pedestrians go out so that the outflow rate does not influence the
flow through the door. We assume pedestrians cannot pass through walls,
but can move along them: we impose free-slip boundary conditions
(15)
~v · ~n = 0,
∂ρ
=0
∂n
at Γw .
In order to implement (15) in the case of the second order model we
compute the fluxes F(Ui , Ug , ~n) through boundary facets using an interior
state Ui and a corresponding ghost state Ug . In particular, we choose
(16)
ρg = ρi ,
~vg = ~vi − 2(~vi · ~n)~n
at wall boundary Γw and ρg = 0.1ρmax , ~vg = vmax~n for the outflow Γo .
Remark 3.1. For the numerical flux function F(Ui , Ug , ~n) we use the HLL
approximate Riemann solver [17]. However, our numerical simulations show
that the condition (15) is not satisfied at the wall boundary. Sub iterations
would be needed at each time step to converge to the correct solution.To
reduce the computational cost, after computing U ∗ in (13) we set to zero at
wall boundary nodes the component of the velocity normal to the boundary.
Adding the source term preserves the slip-wall boundary condition.
Remark 3.2. (Mass conservation) It is essential that there is no loss of
the mass through the wall boundary during numerical simulation. The HLL
solver [17] with the ghost state defined by (16) satisfies this condition when
(15) holds. In fact, let us consider four possible combinations of minimum
and maximum wave speeds (11). Using (16) we get vgn = −vin = 0 at the wall
boundary, where the last equalityp
is due to Remark 3.1. Then we always have
(σi , σg ) = (−si , si ), where si = P 0 (ρi ). Therefore the flux F(Ui , Ug , ~n) is
always in the center region of the HLL solver, that is σL ≤ 0 ≤ σR and its
first component is zero if Ug is defined by (16).
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
PARAMETER NAME
SYMBOL
desired speed
relaxation time
maximal density
pressure coefficient
adiabatic exponent
density-speed coefficient
vmax
τ
ρmax
p0
γ
α
VALUE
9
UNITS
1−7
m/s
0.61
s
7
ped/m2
0.005 − 10 ped1−γ · m2+γ /s2
2−5
7.5
-
Table 1. Parameters values used in the simulation
The first order model consists only of the mass conservation equation
and the boundary conditions are imposed by defining directly the fluxes at
boundary facets. More precisely, we set F = 0 at Γw and F = ρmax V (ρmax )
at Γo .
In our simulation we consider initial conditions of the form ρ0 (x) = ρ̄
in Ω0 and ~v0 (x) = 0, where ρ̄ is a positive constant and Ω0 is an area
inside the evacuation domain far from the exit, see Fig 1. This means that
all pedestrians are placed inside the evacuation room and start to move at
t = 0.
4. Numerical results
In this section we explore numerically the evacuation dynamics of
pedestrians from a room through a narrow exit using the second order model
(1) and the numerical scheme presented in the previous section. We analyze
the dependence of its solutions on some of the parameters of the system
such as the pressure coefficient p0 , the adiabatic exponent γ and the desired
speeds vmax . Finally, we study the effect of obstacles on the evacuation from
a room through a narrow exit. This analysis is preceded by a numerical error
analysis of the scheme and a comparison between the first (5) and the second
(1) order model. More details can be found in [54].
In numerical simulations we use the parameters listed in Table 4. Maximal velocity vmax , maximal density ρmax and the response time τ are chosen from the available literature on experimental studies of pedestrian behaviour, (see [7, 51]). The values of some of the parameters of model (1),
such as p0 , γ, α to authors’ knowledge do not have a direct correspondence
with the microscopic characteristics of pedestrian motion.
We introduceZtwo functionals to analyze the results of simulations: the
total mass M (t) =
ρ(x, t)dx of pedestrians inside the evacuation domain
Ω
10
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
Z
∞
M (t)dt. We use the discrete defini-
and total evacuation time Tevac =
0
tions so the total mass at time step tn is approximated by M n =
and the discrete total evacuation time by Tevac =
∞
X
N
X
ρni |Ci |
i=1
M n ∆tn .
n=1
4.1. Error analysis. In this section we analyze the accuracy of the numerical scheme presented in Section 3. More precisely, we first estimate
convergence order of the method used to solve the eikonal equation and to
compute its gradient. Then we perform the analysis for the fully discrete
scheme for the second order model. Let Ωk be a mesh with Nk finite volume
cells and δi be a surface area of the finite volume cell associated with the
i-th vertex of the mesh Ωk . We consider the L1 error between the reference
solution uref and the approximated one uhk in the form
(17)
Ek =
Nk
X
|uihk − uiref |δi .
i=1
We assume that
Eki = uihk − uiref = Chpk + h.o.t,
C − constant.
p
The grid-spacing parameter hk has the form hk = Nref /Nk , where Nref is
the number of finite volume cells for the reference mesh (see [47]). When an
explicit, analytic solution is available we replace Nref with the area of the
domain |Ω|. We estimate the order of convergence p using the least square
method applied to the logarithm of the equation (18) with neglected higher
order terms.
(18)
4.1.1. Eikonal equation and gradient calculator. We are going to estimate
the order of the algorithm presented in the previous section to solve the
eikonal equation and to compute the gradient of its solution. In the following
tests we use the running cost (8) with the speed-density relation given by
(3) and vmax = 2 m/s and ρmax = 7 ped/m2 . Two cases are analyzed.
Test 1: We consider a domain Ω = [0, 2 m]×[0, 0.2 m] with an outflow
localized at x = 0 and density distribution ρ(x) = x · χ[0,0.5) + 1 · χ[0.5,1) +
(x + 1) · χ[1,1.5) + 2.5 · χ[1.5,2] ped/m2 , where χA is zero outside the set A,
see Fig 2. The solution to the eikonal equation (7) and its gradient can be
found explicitly from
Z x
1
1
~
dx,
and
∇φ =
,0 .
φ(x) =
V (ρ(x))
0 V (ρ)
Test 2: We analyze a domain Ω = [0, 10 m] × [0, 6 m] with an exit
of width L = 1 m centered at (x, y) = (10, 3) and five columns of radius
r = 0.22 m in its interior, see Fig 3. The density is set to zero everywhere
and the reference solution is obtained on a very fine grid with N = 136507
finite volume cells.
In Fig 4 we present the loglog plot of the dependence of the L1 error
on the grid-spacings hk for the eikonal function and the gradient. In Table 2
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
11
Figure 2. Density distribution (in red) and the explicit solution
to the eikonal problem (7), (8) with Γ0 = {(x, y) : x = 0, y ∈
[0, 0.2]} (in blue) as functions of the x-variable in Ω = [0, 2 m] ×
[0, 0.2 m]
we present the corresponding estimates of the convergence order, obtained
using the least square method. We observe that the values of the orders are
close to one. In particular, gradient is computed with a higher order method
and its computation does not decrease the order of the full scheme.
Figure 3. A room 10 m × 6 m with a 1 m wide, symmetrically placed exit and columns with radius r = 0.23 m centered at
(9.5, 2), (9.0, 2.5), (8.5, 3), (9.0, 3.5), (9.5, 4).
Test 1
Test 2
Figure 4. L1 errors of the eikonal solver and the gradient (14)
as functions of the grid-spacing h in the log-log scale for Test 1: a
long corridor with a discontinuous density (on the left) and Test
2: a room with five columns (on the right)
12
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
Test 1
Test 2
Eikonal func. Gradient-x Gradient-y
1.063
1.012
−
0.923
0.903
0.881
Estimates of the convergence order of the eikonal
solver presented in Section 3.3 and the method to compute a gradient (14).
Table 2.
4.1.2. Second order model. Now we perform the same error analysis for the
fully discrete scheme (13) for the second order model (1). We consider the
same domain as in Test 2 in the previous section, but with initial data
ρ0 = 1 ped/m2 in Ω0 = [1 m, 5 m] × [1 m, 5 m], ~v0 = 0 m/s and parameters
vmax = 2 m/s, ρmax = 7 ped/m2 , p0 = 0.005, γ = 2.
The L1 error is computed using (17) with the reference mesh containing Nref = 70772 finite volume cells. In Fig 5 we present the dependence
at time t = 5 s of the L1 error on the number of finite volume cells (on the
left) and its loglog plot with respect to the grid-spacing parameter hk . Least
square method gives the following estimates on the order of the full scheme:
0.8 for the density, 1.14 and 1.05 for the velocity in the x and y direction
respectively.
Figure 5. L1 errors of the density and the velocity for the second
order model (1) as a function of the number N of finite volume cells
(on the left) and as a function of the grid-spacing h in the log-log
scale (on the right).
4.2. Comparison between the first and the second order models.
Now we compare numerically the behaviour of Hughes model (5) and the
second order model (1). We analyze the capability of the models of reproducing the formation of stop-and-go waves and we study the effect on the
flow of obstacles placed in the proximity of an exit.
4.2.1. Stop and go waves. In high density crowd pedestrians experience
strong local interactions which can result in macroscopically observed phenomena. One of them, known from vehicular traffic flow, are stop-and-go
waves. This corresponds to regions with high density and small speed which
propagate backward the flow. In the case of pedestrians such waves were
observed in front of the entrance to the Jamarat Bridge on 12 January 2006
[24] and studied experimentally for a single lane [49, 38].
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
13
Simulating stop-and-go waves can be one of the criteria to validate
a model, see for example [38] in case of microscopic description. In order
to verify if macroscopic models are able to reproduce this phenomenon we
consider a corridor 100 m × 20 m with two, 1.2 m wide exits centered symmetrically at x = 67 m and x = 93 m, see Fig 6. The initial density is
0
10
20
30
40
50
60
70
80
90
100
20
20
10
10
0
0
10
20
30
40
50
60
70
80
90
0
100
Figure 6. A corridor 100 m × 20 m with two, 1.2 m wide exits
centered symmetrically at x = 67 m and x = 93 m.
ρ0 = 3 ped/m2 in Ω0 = [0, 50 m] × [6 m, 26 m] and the initial velocity is set
to zero. Fig 7 shows the density distribution at different times t = 30, 40, 60
s in four cases: the first order model (5) with the running cost function
c(ρ) = 1/vmax (I) and c(ρ) = 1/V (ρ) (II), the second order model (1) with
c(ρ) = 1/V (ρ) and p0 = 0.1 (III), p0 = 0.005 (IV). Other parameters are
vmax = 2 m/s, ρmax = 10 ped/m2 , γ = 2. We observe that in case (I)
the alternative, further exit is not used by pedestrians, who choose their
route only on the basis of the shortest distance, not the shortest time, to
the target. The distribution of density given by the first order model (I, II),
apart from the proximity of the exit, is very smooth. A similar behaviour is
obtained for the second order model with large pressure coefficient p0 = 0.1
(III). High internal repealing forces between pedestrians prevent from congestion and formation of significant inhomogeneities in the density distribution. Decreasing p0 allows for smaller distances between individuals, which
causes stronger interactions. As a result for p0 = 0.005 (IV) we observe
sub-domains with much higher density. The locations of high density peaks
are moving in the opposite direction to the flow. This propagation is clearly
observed near the exit where characteristic arching appears. Fig 8 shows the
density distribution profile along the y-direction originating at the center of
one of the exits. Stop-and-go waves start at the exit and move backwards.
4.2.2. Bottlenecks. It was experimentally observed that flow of pedestrians
through a bottleneck depends on its width (see for example [46, 35, 26]) and
can be significantly slowed down due to clogging at its entrance. Blocking
at bottlenecks occurs when the flow of pedestrians towards the door is much
higher than the capacity of the exit. The density grows and, as a result,
physical interactions between pedestrians increase, slowing down the motion
and interrupting the outflow. From the point of view of evacuation strategies
it is essential for a mathematical model to be able to capture this effect. We
study numerically the behaviour of solutions of the first (5) and the second
(1) order model during the evacuation through a narrow exit. We consider a
room 10 m × 6 m with a 1 m wide, a symmetrically placed exit and different
14
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
I
II
III
IV
Figure 7. Density profiles at different times t=30 s (left), t=40
s (middle), t=60 s (right). I: First order model (5) with c(ρ) = 1/
vmax (Simple model) II: First order model (5) with c(ρ) = 1/V (ρ)
(Hughes model) III: Second order model with c(ρ) = 1/V (ρ) and
p0 = 0.1 IV: Second order model with c(ρ) = 1/V (ρ) and p0 =
2
0.005. Other parameters are vmax = 2 m/s, ρmax = 7 ped/m , γ =
2
2 and initial data are ρ0 = 3 ped/m in Ω0 = [0, 50 m]×[6 m, 26 m]
and ~v0 = 0 m/s.
Figure 8. Time evolution of a density profile along a line parallel
to the y-axis and passing the center of the first exit (x, y) = (64, 6).
obstacles placed in its interior, see Fig 9,
obstacle 1: one column with radius r = 0.3 m,
obstacle 2: three columns with radius r = 0.2 m,
obstacle 3: two walls.
Fig 10 presents the time evolution of the total mass of pedestrians
remaining inside the room for Hughes’ model (5) (on the left) and for the
second order model (1) (on the right) with vmax = 2 m/s, ρmax = 7 ped/m2 ,
P (ρ) = 0.005ρ2 and with initial data ρ0 = 1 ped/m2 in Ω0 = [1 m, 5 m] ×
[1 m, 5 m]. We observe that in the case of the first order model the total
mass M (t) decreases linearly and is the same for all obstacles and the empty
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
Obstacle 1
Obstacle2
15
Obstacle3
Figure 9. A room of dimensions 10 m × 6 m with a 1 m wide,
symmetrically placed exit and different obstacles placed in its interior. Obstacle 1: circle centered at (8.5, 3) with radius r = 0.3 m.
Obstacle 2: 3 circles centered at (9, 2.5), (8, 3), (9, 3.5) with radius
r = 0.2 m. Obstacle 3: 2 rectangles with 7.5 m ≤ x ≤ 9 m and
2.3 m ≤ y ≤ 2.5 m, 3.5 m ≤ y ≤ 3.7 m.
room. The outflow is regulated only by the capacity of the door. We do
not observe either clogging, which would slow down the decrease of the
total mass, or the influence of obstacles. The total evacuation time Tevac
is basically the same for all situations. On the other hand, in the case of
the second order model the empty room experiences a significant decrease
of the outflow. It may correspond to clogging at the exit when a sufficiently
large number of pedestrians reaches it. Obstacles play a role of a barrier
and decrease the flow arriving at the exit. As a result, evacuation is slower
but clogging is reduced.
Hughes model
Second order model
Figure 10. Time evolution of the total mass M (t) of pedestrians
in an empty room and in a room with three different obstacles for
2
the Hughes model (5) with vmax = 2 m/s, ρmax = 7 ped/m (on
the left) and for the second order model (1) with vmax = 2 m/s,
2
ρmax = 7 ped/m , p0 = 0.005, γ = 2 (on the right).
4.3. Room evacuation. We now focus our analysis on the second order
model (1) and explore the dependence of its solutions on the system parameters p0 , γ and vmax . Our aim is to analyze how the degree of congestion of
the crowd and the value of desired walking speed of pedestrians influence
the evacuation time.
4.3.1. Dependence on the parameters p0 and γ. At first we analyze the dependence on the coefficients of the internal pressure function given by the law
for isentropic gases (4). It describes the repealing forces between pedestrians and prevents from overcrowding. Under emergency and panic conditions
16
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
the comfort zone of pedestrians, which defines how close they can stay to
each other, decreases. As a result, the density of the crowd can increase and
impedes the movement leading to discontinuous flow. As we have already
seen, in Section (4.2.1), the formation of stop-and-go waves depends on the
coefficient p0 . Now our aim is to study the relation between the strength of
the repealing forces between pedestrians and the efficiency of the evacuation.
We consider an empty room 10 m×6 m with a 1 m wide, symmetrically
placed exit and set the following initial data ρ0 = 1.5 ped/m2 in Ω0 =
[1 m, 5 m]×[1 m, 5 m], ~v0 = 0 m/s. In Fig 11a we present the time evolution
of the total mass of pedestrians remaining in the room for different values of
the coefficient p0 in the cases of adiabatic exponent γ = 2 and vmax = 2 m/s,
ρmax = 7 ped/m2 . We observe that for small values of p0 there is a significant
decrease of the outflow of pedestrians through the exit, which means that it is
blocked. Increasing p0 prevents from congestion and as a result the outflow is
smother. However, when the distance between pedestrians increases, leaving
the room becomes more time consuming. Fig 12 shows the dependence of
the total evacuation time Tevac on the parameter p0 in the case of γ = 2, 3.
We observe an optimal value of the parameter p0 ∼ 0.5, which minimizes
the evacuation time of pedestrians from the room.
The effect of the adiabatic exponent γ is similar. High values increase
the repealing forces between pedestrians. We present in Fig 11b the evolution of the total mass for γ = 2, 3, 4, 5. Clogging observed in the case γ = 2
diminishes for larger γ. For γ = 5 we observe quasi-linear decrease of the
total mass.
The response of pedestrians to compression has an essential effect on
the evacuation time. In normal conditions, when pedestrians want to keep
a certain comfort and have enough free space to move, the outflow through
the exit is undisturbed. During emergency situations the distances between
individuals decrease and the density of the crowd increases, reducing the
mobility of pedestrians. As a result the flow may become discontinuous and
exits may be blocked. Our simulations show that the second order model is
able to reproduce these phenomena.
4.3.2. Dependence on the desired speed vmax . Now we look for the dependence of the total evacuation time on the desired walking speed vmax of
pedestrians. Empirical studies indicate that the average free speed, that is
the speed at which pedestrians walk when they are not influenced by others, is about 1.34 m/s with standard deviation of 0.37 m/s (see [7]). Due
to impatience, emergency or panic, people tend to move faster to escape
uncomfortable situation or direct life thread as soon as possible. Using the
social force model [25] Helbing et al. [19] analyzed an evacuation of 200
people from a room for different desired speeds corresponding to different
states of panic. Under the condition of high friction due to the tangential
motion of pedestrians, they observed the existence of an optimal speed for
which the evacuation was optimized. In Fig 13 we analyze the evacuation
of pedestrians from the empty room Ω = [0, 10 m] × [0, 6 m] through a 1 m
wide exit with ρ0 = 2 ped/m2 in Ω0 = [1 m, 5 m] × [1 m, 5 m], ~v0 = 0 m/s
for different values of the desired speed vmax = 0.5, 1, 1.5, 2, 3, 4, 6 m/s in the
case of p0 = 0.005 (on the left) and p0 = 0.5 (on the right).
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
17
Figure 11. Time evolution of the total mass M (t) of pedestrians
inside a room without obstacles for the second order model (1)
2
with vmax = 2 m/s, ρmax = 7 ped/m γ = 2 for different pressure
−3
−2
coefficients p0 = {5×10 , 10 , 5×10−2 , 10−1 , 5×10−1 , 1, 2, 5, 10}
2
(on the left) and with vmax = 2 m/s, ρmax = 7 ped/m , p0 =
5 × 10−3 for different adiabatic exponents γ = 2, 3, 4, 5 (on the
2
right). The initial density is ρ0 = 1.5 ped/m in Ω0 = [1 m, 5 m] ×
[1 m, 5 m], and the initial velocity ~v0 = 0 m/s.
Figure 12. Total evacuation time Tevac for the second order
2
model (1) with vmax = 2 m/s, ρmax = 7 ped/m , γ = 2, 3 as
a function of the pressure coefficient p0 = {5 × 10−3 , 10−2 , 5 ×
10−2 , 10−1 , 2.5×10−1 , 5×10−1 , 7.5×10−1 , 1, 1.25, 1.5, 2, 5, 10}. The
2
initial density is ρ0 = 1.5 ped/m in Ω0 = [1 m, 5 m] × [1 m, 5 m],
and the initial velocity ~v0 = 0 m/s.
We observe that unlike Helbing et al. [19], the total evacuation time
decreases with higher desired speed. Pedestrians at the front of a group can
move almost at their desired velocity vmax as they are not slowed down by
the presence of others. When the free speed is high, they reach the exit very
fast and leave the room in a short time. However, due to the limited flow
capacity of the exit, in the case of p0 = 0.005 we see that pedestrians start
to accumulate in front of the door and block it. Decreasing the value of vmax
the flow through the exit decreases, so clogging occurs earlier than in the
case of larger desired speed. At the same time, also the accumulation has a
smaller rate, due to smaller vmax , so the outflow is only slowed down instead
of being blocked. Increasing the value of the internal pressure coefficient
18
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
Figure 13. Time evolution of the total mass M (t) of pedestrians
inside a room without obstacles for the second order model (1) with
2
ρmax = 7 ped/m , γ = 2 and different values of the desired velocity
vmax = {0.5, 1, 1.5, 2, 3, 4, 6} m/s and different pressure coefficients
p0 = 0.005 (on the left) and p0 = 0.5 (on the right). The initial
2
density is ρ0 = 5 ped/m in Ω0 = [1 m, 5 m] × [1 m, 5 m], and the
initial velocity ~v0 = 0 m/s.
to p0 = 0.5, the evacuation becomes faster and more regular, as we have
already observed in the previous simulations.
4.3.3. Effect of obstacles. In this section we study the evacuation from a
room following the idea of Hughes [30], who raised the question of weather
suitably placed obstacles can increase the flow through an exit. This idea
is an inversion of the Braess paradox [4, 5], which was formulated for traffic
flows and states that adding extra capacity to a network can in some cases
reduce the overall performance. In the case of crowd dynamics, placing an
obstacle may be seen intuitively as a worse condition. Nevertheless, it is
expected to decrease the density in front of the exit and as a result prevent
it from blocking.
We examine closely this phenomenon in case of pedestrian motion
using the second order model (1) with vmax = 2 m/s, ρmax = 7 ped/m2 . In
Section 4.2.2 we have already observed that an obstacle in front of an exit
can reduce the clogging. In this section we consider a room 10 m × 6 m with
a 1 m wide, symmetrically placed exit and system of five circular columns
arranged in the shape of a triangle opened towards the door. The columns
are centered at (9.5, 2), (9, 2.5), (8.5, 3), (9, 3.5), (9.5, 4) and have the radius
r = 0.22 m, see Fig 3 and the initial density equals ρ0 = 1 ped/m2 in
Ω0 = [1 m, 5 m] × [1 m, 5 m].
In order to compare the efficiency of the evacuation we use the total
mass of pedestrians that remain in the room, which corresponds directly
to the outflow through the door. Fig 14 presents the time evolution of the
total mass on meshes with different number of finite volume cells, which
are refined by a factor two near the exit and the obstacles. The room
with the columns and the obstacle-free case are considered. We observe
that for the empty room (on the left) the results do not change with the
mesh refinement. On the contrary, in the case with the obstacles (on the
right) there is a significant difference between the total mass curves for
N = 16000, 26000 and for N = 6000, 8000 finite volume cells. This effect is
a result of the presence of phenomena occurring at different length scales for
MACROSCOPIC MODELING AND SIMULATIONS OF ROOM EVACUATION
19
irregular flows. In Fig 15 we can see that the clogging observed on the mesh
with N = 16000 finite volume cells is not resolved for N = 6000 resulting in
false faster evacuation.
Figure 14. Time evolution of the total mass M (t) of pedestrians
for the second order model (1) with vmax = 2 m/s, ρmax = 7
2
ped/m , p0 = 0.005, γ = 2 for different number N of finite volume
cells in case of an empty room (on the left) and a room with five
columns in front of the exit (on the right).
In the following test we compare numerically the evacuation from the
room described above using a mesh with N = 16000 finite volume cells.
Fig 16 shows the time evolution of the total mass for two different pressure
coefficients p0 = 0.005, 0.001. We observe that the clogging present in the
empty room is reduced significantly using the obstacles. The system of
columns creates a ”waiting zone” in front of the exit. Pedestrians are slowed
down and partially stopped by the obstacles, which corresponds to the slower
outflow in the initial phase of the evacuation. But, at the same time the
density at the exit remains low because the incoming flow does not exceed
the door capacity. In case of small pressure p0 = 0.001, which allows for
higher congestion, the improvement due to the obstacles is more visible.
For the initial mass M = 32 and columns of radius r = 0.24 m the total
evacuation time of the room with the columns becomes smaller that in the
obstacles-free situation.
fine mesh
coarse mesh
Figure 15. Density profiles at time T = 9 s for the second order
2
model (1) with vmax = 2 m/s, ρmax = 7 ped/m , p0 = 0.005, γ = 2
for different numbers of the finite volume cells: N = 16000 (on the
left) and N = 6000 (on the right).
20
M. TWAROGOWSKA, P. GOATIN, R. DUVIGNEAU
Figure 16. Time evolution of the total mass M (t) of pedestrians
for the second order model (1) with vmax = 2 m/s, ρmax = 7
2
ped/m , γ = 2 and different pressure coefficients: p0 = 0.005 (on
the left) and p0 = 0.001 (on the right).
5. Conclusions
In this study, two pedestrian flow models have been analyzed in the
context of macroscopic modelling of evacuation of pedestrians from a room
through a narrow exit. Error analysis has been carried out for numerical
validation of a finite-volume scheme on unstructured grid.Various test-cases
have been considered to quantify the influence of the model parameters
on the behaviour of solutions and to measure the ability of the models to
reproduce some of the phenomena occurring in the evacuation of high density crowds. More precisely, numerical experiments show that the classical
Hughes’ type model cannot reproduce stop-and-go waves or clogging at bottlenecks. On the other hand, it was verified numerically that the second
order model captures better the structure of interactions between pedestrians and is able to produce the above behaviours. However, even this model
is still far from being validated and should be verified and calibrated with
realistic experiments. In fact, we have pointed out that values of some of its
parameters have a significant effect on the formation of the above phenomena so their tuning is essential. For some particular choices of the parameters
the evacuation through a narrow exit was analyzed and an example of the
inverse Braess paradox was given. It was shown that using a particular configuration of obstacles it is possible to reduce the clogging at the exit and
increase the outflow. Analysis of more realistic settings are to be considered
at the next step.
Acknowledgment
This research was supported by the European Research Council under
the European Union’s Seventh Framework Program (FP/2007-2013) / ERC
Grant Agreement n. 257661.
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